A morphism locally of finite presentation is étale exactly when it is smooth of relative dimension \(0\), and exactly when it is flat and unramified.
Unramified and étale
Unramified morphisms (Tag D-MORUNR)
\(f : X \to Y\) locally of finite type is unramified if for every \(x \in X\) with \(y = f(x)\), \[ f^\sharp(\mfm_y) \OO_{X,x} = \mfm_x \qquad\text{and}\qquad \kappa(x)/\kappa(y) \text{ is a finite separable extension.} \] Equivalently, \(\Omega_{X/Y} = 0\); equivalently, \(\Delta_{X/Y}\) is an open immersion.
The first condition says that for every \(y\in Y\) and \(x\in X_y\), the local ring \(\OO_{X_y,x}\) is the field \(\kappa(x)\), a finite separable extension of \(\kappa(y)\). Since the diagonal of an unramified morphism is open, two sections \(s,t\colon Y\to X\) of an unramified morphism \(f\) agree on an open subset of \(Y\), namely \((s,t)^{-1}(\Delta_{X/Y})\).
For a nonconstant map of smooth integral curves over an algebraically closed field, the local picture at a closed point is \(f^\sharp(\mfm_{f(p)}) \OO_{X,p} = \mfm_p^{e_p}\), and \(f\) is unramified exactly when every ramification index \(e_p\) equals \(1\). For \(n>1\), \(t \mapsto t^n\) on \(\AA^1\) is ramified at the origin. In characteristic \(p\) dividing \(n\), its differential vanishes everywhere and the induced function-field extension is inseparable. For an unramified finite separable morphism \(f\colon X\to Y\) of smooth projective curves, the ramification divisor is \(0\), and Riemann–Hurwitz gives \(2g_X-2=(\deg f)(2g_Y-2)\).
Étale morphisms (Tag D-MORETALE)
\(f : X \to Y\) is étale if it is flat and unramified, equivalently flat, locally of finite presentation, with \(\Omega_{X/Y} = 0\), equivalently smooth of relative dimension \(0\).
An étale morphism is open. Finite étale morphisms form the category of covers used to define the étale fundamental group.
Étale does not imply local isomorphism in the Zariski topology: \(\GG_m \to \GG_m\), \(t \mapsto t^2\), over a field of characteristic not \(2\) is finite étale of degree \(2\) and is not an isomorphism over any nonempty Zariski open. Its base change along itself is trivial: \(\GG_m\times_{\GG_m}\GG_m\cong\GG_m\sqcup\GG_m\), the two components corresponding to the two square roots of \(1\).
A finite étale morphism of degree \(1\) is an isomorphism. Bijectivity on underlying points does not suffice: \(\Spec L\to\Spec k\) is bijective and finite étale for a nontrivial finite separable field extension \(L/k\).
A morphism locally of finite type is unramified exactly when \(\Omega_{X/Y}=0\), and étale exactly when it is also flat. A finite morphism of smooth curves over an algebraically closed field is unramified at \(p\) exactly when \(e_p=1\); these are the ramification indices in Riemann--Hurwitz.
Ramification index and ramification divisor
Ramification index, tame and wild ramification (Tag D-IV2RAM)
Let \(f : X \to Y\) be a finite morphism of curves, \(p \in X\), \(q = f(p)\), and \(t\) a uniformizer of the discrete valuation ring \(\OO_{Y,q}\). The ramification index is \[ e_p \definedas v_p(f^\sharp t) , \] the valuation at \(p\) of the pulled-back uniformizer. \(f\) is ramified at \(p\) if \(e_p > 1\) and unramified there if \(e_p = 1\); the image \(q\) of a ramification point is a branch point.
(Pullback of divisors)
For \(f : X \to Y\) a finite morphism of nonsingular curves, \(f^* : \operatorname{Div} Y \to \operatorname{Div} X\) is the homomorphism with \[ f^* q = \sum_{p \in f^{-1}(q)} e_p \, p . \] It preserves linear equivalence and satisfies \(f^* \mcl(D) \cong \mcl(f^* D)\), so it induces \(f^* : \Pic Y \to \Pic X\), and \(\deg f^* D = \deg f \cdot \deg D\). [@Har10a, §IV.2, Proposition II.6.9, Exercise II.6.8]
(Ramification divisor and branch locus)
For \(f \colon X \to Y\) a finite separable morphism of smooth projective curves, the ramification divisor is \[ R = \sum_{p \in X} \length (\Omega_{X/Y})_p \cdot p , \] which is \(\sum_p (e_p - 1)\, p\) when all ramification is tame. The branch locus is the finite set \(f(\supp R)\) of branch points, and the branch divisor is the pushforward \(f_* R = \sum_p \length(\Omega_{X/Y})_p \cdot f(p)\).
(Tame and wild)
Let \(\characteristic k = p_0\). The ramification at \(p\) is tame if \(p_0 \nmid e_p\), and wild if \(p_0 \mid e_p\). In characteristic \(0\) all ramification is tame.
(The length formula)
For \(f\) finite separable, \(\Omega_{X/Y}\) is a torsion sheaf supported exactly on the ramification locus, its stalks are principal \(\OO_p\)-modules, and \[ \length (\Omega_{X/Y})_p \begin{cases} = e_p - 1 & \text{at a tamely ramified } p, \\ > e_p - 1 & \text{at a wildly ramified } p . \end{cases} \]
(Where the dichotomy comes from)
The stalk length is computed by one derivative. With \(u\) a uniformizer at \(p\) and \(t\) one at \(q\), write \(f^\sharp t = c u^{e_p}(1 + \cdots)\); then \[ f^* dt = \left( e_p \, c \, u^{e_p - 1} + \cdots \right) du , \] and \(\length(\Omega_{X/Y})_p\) is the valuation of that coefficient. If \(p_0 \nmid e_p\) the leading term survives and the valuation is \(e_p - 1\). If \(p_0 \mid e_p\) the leading term is killed, the valuation is determined by whatever comes next, and it is strictly larger. So the ramification at \(p\) is tame if and only if \(e_p\) is invertible in \(k\).
The map \(u \mapsto t=u^n\) with \(p_0 \nmid n\) is tamely ramified at \(u=0\), where \(R\) has coefficient \(n-1\). For a wild example take \(\characteristic k = p_0\) and \(f : \PP^1 \to \PP^1\) with \(t = u^{p_0} - u\), which is separable since \(dt = -du\). It is unramified on the affine line and totally ramified at infinity with \(e = p_0\), which is wild, and the local computation there gives length \(2p_0 - 2 > p_0 - 1\). Riemann–Hurwitz confirms it: \(-2 = p_0 \cdot (-2) + \deg R\) forces \(\deg R = 2p_0 - 2\), carried by that one point.
(Ramification and Riemann--Hurwitz)
Riemann–Hurwitz (T-LKT0U) holds with \(\deg R = \sum_p \length (\Omega_{X/Y})_p\), which equals \(\sum_p (e_p - 1)\) when all ramification is tame. At a wildly ramified point the local length exceeds \(e_p-1\), so \(\deg R>\sum_p(e_p-1)\). For the Artin–Schreier map \(u\mapsto u^{p_0}-u\) of degree \(p_0\), the tame formula would give \(\deg R=p_0-1\) at its single ramification point, while Riemann–Hurwitz requires \(\deg R=2p_0-2\).
The condition \(e_p = 1\) at every \(p\) is unramifiedness in the sense of D-MORUNR; a nonconstant morphism of smooth curves is flat, so it is étale if and only if \(e_p=1\) at every \(p\).
For a uniformizer \(t\) at \(f(p)\), \(e_p=v_p(f^*t)\), and the ramification divisor is \(R=\sum_p\operatorname{length}(\Omega_{X/Y})_p\cdot p\). The length equals \(e_p-1\) exactly when \(e_p\) is invertible in \(k\); in the wild case it is strictly larger than \(e_p-1\).
The ramification divisor has even degree (Tag PR-IV2DEGREVEN)
Let \(f : X \to Y\) be a finite separable morphism of curves. Then \(\deg R\) is even.
\(R \sim K_X - f^* K_Y\), so \[ \deg R = (2 g_X - 2) - \deg(f)(2 g_Y - 2) , \] a difference of two even integers.
(Parity consequences)
The proof is Riemann–Hurwitz with the terms moved, and the resulting parity constraint gives a consistency check on any ramification count. A branching pattern with odd \(\deg R\) is impossible, so parity gives an immediate consistency check on ramification counts.
Used forward with tame simple ramification, where every \(e_p = 2\) and \(\deg R\) counts the branch points, it says such a cover has an even number of branch points. This is the reason a degree-\(2\) cover of \(\PP^1\) is branched at \(2g+2\) points rather than an arbitrary number, and the reason the Legendre normal form of an elliptic curve has exactly four branch points and not three. It also rules out possibilities directly: no simply branched cover of \(\PP^1\) has an odd branch count, whatever its degree.
The hypothesis that bites is separability, since the whole statement descends from \(K_X \sim f^* K_Y + R\), which needs the cotangent sequence to be short exact. In characteristic \(p\) there is no parity statement about an inseparable morphism because there is no \(R\); see PR-IV2INSEP.
For a degree-two cover of \(\PP^1\) by a curve of genus \(g\), in characteristic \(\neq2\), every ramification point has \(e_p=2\), and parity with Riemann–Hurwitz gives \(2g+2\) branch points.
Étale covers
Étale covers, trivial covers, and simple connectedness of $\PP^1$ (Tag D-IV2ETCOV)
A morphism \(f : X \to Y\) is an étale cover if it is finite and étale. It is a trivial cover if \(X \cong \coprod_{i \in I} Y\) for a finite index set \(I\), with \(f\) the identity on each copy. \(Y\) is simply connected if every étale cover of \(Y\) is trivial, equivalently \(\pi_1^{\Et}(Y) = 0\).
\(\PP^1_k\) over \(k = \kbar\) is simply connected, in every characteristic.
Let \(f : X \to \PP^1\) be an étale cover; it is enough to treat \(X\) connected, and then to show \(f\) is an isomorphism. \(X\) is regular because \(f\) is étale and \(\PP^1\) is smooth over \(k\), and a connected regular scheme is irreducible, so \(X\) is a curve; \(f\) is finite, so \(X\) is projective; \(f\) is unramified, so \(k(X)/k(\PP^1)\) is separable. Riemann–Hurwitz therefore applies, and \(R = 0\) since \(f\) is unramified, so with \(n = \deg f\), \[ 2 g_X - 2 = n(2 \cdot 0 - 2) = -2n . \] Then \(g_X \geq 0\) gives \(-2n \geq -2\), so \(n = 1\) and \(g_X = 0\), and a finite morphism of degree \(1\) between curves is an isomorphism.
(Topological analogue)
For a connected covering space \(S\to S^2\) of degree \(n\), \(\chi(S)=n\chi(S^2)=2n\), and \(\chi(S)\le2\) for a compact connected surface, so \(n=1\); the Riemann–Hurwitz proof for \(\PP^1\) replaces \(\chi\) by \(2-2g\) and the covering space by a finite étale morphism.
(The affine line in characteristic $p$)
Let \(\operatorname{char}k=p>0\). The Artin–Schreier cover \(\Spec k[x,y]/(y^p - y - x)\to\AA^1\), \((x,y)\mapsto x\), is finite étale of degree \(p\) and connected, because \(d(y^p - y) = -dy\) never vanishes; so \(\AA^1_k\) is not simply connected. The cover extends to a morphism \(\PP^1\to\PP^1\) that is totally and wildly ramified at \(\infty\) (D-IV2RAM). Over \(\CC\), \(\AA^1\) has no nontrivial finite étale covers.
(Finite étale covers)
Étale morphisms are defined in D-MORETALE. For a proper nonempty open subset \(U\subseteq Y\), the morphism \(U\sqcup Y\to Y\) is étale and surjective but not finite, and it is not of the form \(\coprod_{i\in I}Y\to Y\).
A connected finite étale cover \(X\to\PP^1\) of degree \(n\) has \(2g_X-2=-2n\) by Riemann–Hurwitz, so \(n=1\): \(\PP^1\) has no nontrivial connected finite étale covers. In characteristic \(p\) the affine line has the nontrivial connected finite étale Artin–Schreier covers \(y^p-y=x\).
Smoothness
Smooth morphisms and relative dimension (Tag D-MORSM)
\(f : X \to Y\) is smooth if it is flat, locally of finite presentation, and has geometrically regular fibres. It is smooth of relative dimension \(n\) if in addition every irreducible component of every fibre has dimension \(n\).
(Criteria for smoothness of relative dimension $n$)
For a morphism \(f \colon X \to Y\) the following are equivalent.
\(f\) is smooth of relative dimension \(n\).
\(f\) is flat and locally of finite presentation, every fibre has pure dimension \(n\), and \(\Omega_{X/Y}\) is locally free of rank \(n\).
\(f\) is flat and locally of finite presentation, and every fibre \(X_y\) is smooth of pure dimension \(n\) over \(\kappa(y)\).
\(f\) is flat and locally of finite presentation, and every geometric fibre is a regular scheme of pure dimension \(n\).
Locally on \(X\) and \(Y\), \(f\) is induced by \(B \to B[x_1, \ldots, x_{n+r}]/(g_1, \ldots, g_r)\) with \(\det(\partial g_i / \partial x_j)_{1 \leq i, j \leq r}\) invertible.
The fibre-dimension hypothesis in condition 2 cannot be dropped. Over \(k = \FF_p\), the morphism \(\Spec k[x]/(x^p) \to \Spec k\) is flat and of finite presentation, and \(\Omega = (k[x]/(x^p))\, dx\) is free of rank \(1\) because \(d(x^p) = p x^{p-1} dx = 0\); but the fibre has dimension \(0\), not \(1\), and it is not reduced, so the morphism is not smooth.
Over a non-perfect field a fibre can be regular and not geometrically regular: \(\Spec k[x]/(x^p - t)\to\Spec k\) over \(k = \FF_p(t)\) is flat and of finite presentation, its source is a regular point, and it becomes non-reduced after base change to \(\kbar\); so this morphism is not smooth.
For a smooth morphism of relative dimension \(n\), \(\Omega_{X/Y}\) is the relative cotangent bundle and \(\bigwedge^n\Omega_{X/Y}\) is the relative canonical sheaf. Smooth morphisms are stable under base change and composition. Smooth morphisms of relative dimension \(0\) are étale. A smooth morphism of relative dimension \(n\) locally factors as an étale map to \(\AA^n_Y\) followed by the projection.
Smooth over an algebraically closed field equals regular (Tag T-MORSMREG)
Let \(k\) be a field and \(X\) a scheme locally of finite type over \(k\). If \(X \to \Spec k\) is smooth of relative dimension \(n\) then \(X\) is regular of dimension \(n\). The converse holds when \(k\) is perfect, in particular when \(k = \kbar\), and fails otherwise.
(Jacobian criterion)
Let \(k\) be an algebraically closed field, \(X = V(f_1, \ldots, f_r) \subseteq \AA^m_k\), and \(x \in X\) a closed point with \(\dim \OO_{X,x} = n\). Let \(J(x) = (\partial f_i / \partial x_j (x))\) be the Jacobian matrix at \(x\).
\(\dim_k \mfm_x / \mfm_x^2 = \dim_k \Omega_{X/k} \otimes \kappa(x) = m - \operatorname{rank} J(x)\).
\(X\) is regular at \(x\) if and only if \(\operatorname{rank} J(x) = m - n\).
[@Har10a, Theorem I.5.1, Proposition II.8.7]
(Smoothness over a field)
Let \(k\) be a field and \(X\) a scheme locally of finite type over \(k\), of pure dimension \(n\). The following are equivalent.
\(X\) is smooth over \(k\).
\(\Omega_{X/k}\) is locally free of rank \(n\).
\(X\) is covered by open subschemes isomorphic to open subschemes of \(\Spec k[x_1, \ldots, x_m]/(f_1, \ldots, f_r)\) whose Jacobian matrix has rank \(m - n\) at every point.
\(X_{\bar{k}} = X \times_k \Spec \bar{k}\) is regular.
If \(k\) is perfect, these are also equivalent to: \(X\) is regular.
[@Har10a, Theorem II.8.15, Theorem III.10.2]
Smoothness is a property of a morphism, while regularity is a property of local rings. Smooth is the stronger, base-change-stable notion: it is “regular after every field extension”, which is why the two agree exactly when there are no inseparable extensions to spoil it.
Over an imperfect field the converse fails: for \(k=\FF_p(t)\), \(X=\Spec k[x]/(x^p-t)\) is the spectrum of a field and hence regular, while \(X_{\bar k}=\Spec \bar k[x]/(x-t^{1/p})^p\) is nonreduced, so \(X\) is not smooth over \(k\).
A smooth morphism is regular fibrewise, not absolutely: a smooth morphism over a singular base has singular total space, and the fibres are what the definition controls.
A scheme \(X\) locally of finite type over a field \(k\) is smooth over \(k\) exactly when \(X_{\bar k}\) is regular; for \(k\) perfect this holds exactly when \(X\) is regular. Over the imperfect field \(\FF_p(t)\), \(\Spec\FF_p(t)[x]/(x^p-t)\) is regular, being a field, and is not smooth, since \(\FF_p(t^{1/p})\otimes_{\FF_p(t)}\overline{\FF_p(t)}\) is not reduced.
The jacobian criterion for smoothness (Tag PR-MORJAC)
Let \(k = \kbar\) and let \(X = V(f_1, \dots, f_r) \subseteq \AA^n_k\) be of pure dimension \(n - r\). Then \(X\) is smooth at a closed point \(p\) exactly when the jacobian matrix \[ J(p) = \qty{ \partial f_i / \partial x_j (p) }_{i,j} \] has rank \(r\). In general \(X\) is smooth at \(p\) of dimension \(n - \rank J(p)\) when the two agree, and \(\rank J(p) \leq r\) always, with strict inequality exactly at the singular points.
The criterion is the standard computational tool in the topic: write the equations, differentiate, and solve \(J = 0\) together with the equations themselves.
A point of \(V(J)\) is singular only if it also lies on \(X\).
The hypothesis \(k = \kbar\) is load-bearing and its failure is the same characteristic-\(p\) inseparability seen elsewhere. So is the pure-dimension hypothesis: for \(X = V(xy, xz) \subseteq \AA^3\), a plane meeting a line, the jacobian rank is wrong along the whole line because the codimension is not constant.
The intrinsic statement behind the criterion is that \(\mfm_p/\mfm_p^2\) has dimension \(\dim \OO_{X,p}\), that is, \(\OO_{X,p}\) is a regular local ring; the jacobian just computes the dimension of that cotangent space from a presentation. That equivalence between smooth over \(\kbar\) and regular is what the criterion is really testing.
Where smoothness fails
The Frobenius morphism is nowhere smooth (Tag FE-MORFROB)
Let \(\characteristic k = p\) with \(k = \kbar\) and let \(F : \PP^n_k \to \PP^n_k\) be the Frobenius, given on coordinates by \(x \mapsto x^p\). Then \(F\) is finite, flat, surjective, and a homeomorphism, and it is nowhere smooth. Since \(d(t^p) = 0\), the map \(F^* \Omega_{\PP^n/k} \to \Omega_{\PP^n/k}\) is zero, so \(\Omega_{\PP^n/\PP^n} = \Omega_{\PP^n/k}\) is locally free of rank \(n\) while the relative dimension of \(F\) is \(0\).
This example separates smoothness from flatness and from the topology of the underlying map. \(F\) is finite flat of degree \(p^n\) and bijective on points, so no condition on fibres as sets or on flatness detects the failure; only the differentials do. It is the sharpest illustration that smoothness is not a topological or even a flat-family condition.
It also shows that “étale equals flat plus finite fibres” is false: \(F\) is flat with one point in every fibre and is not étale, because the fibres are the non-reduced \(\Spec k[x]/(x^p)\) rather than reduced points. The separability clause in the definition of unramified is precisely what rules this out.
Generic smoothness (Tag T-MORGEN)
Let \(\characteristic k = 0\) and let \(f : X \to Y\) be a dominant morphism of varieties over \(k\) with \(X\) smooth. Then there is a nonempty open \(V \subseteq Y\) such that \(\ro{f}{f^{-1}(V)} : f^{-1}(V) \to V\) is smooth.
The theorem says that bad behaviour of a morphism is confined to a proper closed subset, so one may always shrink the base and assume smoothness. It is used to define ramification divisors, prove Sard-type statements, and reduce statements about a family to its general member.
Characteristic zero is essential: Frobenius gives a dominant morphism from a smooth variety that is smooth over no nonempty open. The proof in characteristic zero rests on separability of every field extension of \(k(Y)\), which is exactly what fails in characteristic \(p\).
The companion statement, generic flatness, has no characteristic-zero restriction: a finite type morphism to an integral Noetherian base is flat over a nonempty open.
The Frobenius \(F\colon\PP^n_k\to\PP^n_k\) in characteristic \(p\) is a dominant morphism of smooth varieties that is smooth at no point, so generic smoothness fails in characteristic \(p\). \(F\) is finite and flat, and generic flatness holds in every characteristic.
Normal and regular morphisms
Normal and regular morphisms (Tag D-MORNORMREG)
A morphism of locally Noetherian schemes \(f \colon X \to Y\) is normal, respectively regular, if it is flat and for every \(y \in Y\) the fibre \(X_y\) is geometrically normal, respectively geometrically regular, over \(\kappa(y)\): \(X_y \times_{\kappa(y)} \Spec L\) is normal, respectively regular, for every finite field extension \(L/\kappa(y)\). A ring map \(A \to B\) of Noetherian rings is normal or regular when \(\Spec B \to \Spec A\) is.
A morphism locally of finite type is regular exactly when it is smooth.
If \(f \colon X \to Y\) is flat, \(Y\) is regular (respectively normal) and the fibres of \(f\) are regular (respectively normal), then \(X\) is regular (respectively normal).
For a Noetherian local ring \(A\), the completion map \(A \to \hat{A}\) is regular exactly when \(A\) has geometrically regular formal fibres; this holds for every local ring essentially of finite type over a field, and it is part of the definition of an excellent ring.
In characteristic \(p\), the Frobenius \(\Spec \FF_p(t^{1/p}) \to \Spec \FF_p(t)\) is flat and its only fibre is a field, but the fibre is not geometrically reduced: after base change to \(\FF_p(t^{1/p})\) it becomes \(\FF_p(t^{1/p})[x]/(x - t^{1/p})^p\). So this map is not regular, although both schemes are regular.
Henselian rings
Hensel's lemma and henselization (Tag D-HENSEL)
Let \((R, \mfm, k)\) be a complete Noetherian local ring and \(f \in R[x]\) monic. If the reduction \(\bar{f} \in k[x]\) factors as \(\bar{f} = g_0 h_0\) with \(g_0, h_0\) monic and coprime, then \(f = g h\) with \(g, h \in R[x]\) monic, \(\bar{g} = g_0\) and \(\bar{h} = h_0\). In particular a simple root of \(\bar{f}\) in \(k\) lifts to a root of \(f\) in \(R\).
(Henselian ring and henselization)
A local ring \((R, \mfm)\) is henselian if the conclusion of Hensel’s lemma holds for every monic \(f \in R[x]\). The henselization \(R^h\) of a local ring \(R\) is a henselian local ring with a local homomorphism \(R \to R^h\) through which every local homomorphism from \(R\) to a henselian local ring factors uniquely. It is the colimit of the local rings \(\OO_{U,u}\) over étale neighbourhoods \((U, u) \to (\Spec R, \mfm)\) with trivial residue field extension.
Complete local rings are henselian, and \(R \to R^h \to \hat{R}\). For \(R = k[x]_{(x)}\), \(R^h\) is the ring of power series in \(k[[x]]\) that are algebraic over \(k(x)\); it contains \(\sqrt{1+x}\) in characteristic not \(2\), which \(R\) does not, and it does not contain \(e^x\) when \(k = \CC\).